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Motivated by the problem of labeling maps, we i n vestigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows. In On log n time, we can nd an Olog n-factor approximation of the maximum subset in a set of n arbitrary axis-parallel rectangles in the plane. If all rectangles have unit height, we can nd a 2-approximation in On log n time. Extending this result, we obtain a 1 + 1 k-approximation in time On log n + n 2k,1 time, for any i n teger k 1. pairwise disjoint. The label placement problem is to nd a largest feasible con guration.

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For 2D or 3D meshes that represent a continuous function to the reals, the contours-or isosurfaces-of a specified value are an important way to visualize it. To find such contours, a seed set can be used for the starting points from which the traversal of the contours can start. This paper gives the first methods to obtain seed sets that are provably small in size. They are based on a variant of the contour tree (or topographic change tree). We give a new, simple algorithm to compute such a tree in regular and irregular meshes that requires O(rt log n) time in 2D for meshes with n elements, and in 0( n2) time in higher dimensions. The additional storage overhead is proportiid to the maximum size of any contour (linear in the worst case, but typically less). Given the contour tree, a minimum size seed set can be computed in polynomial time and storage. Since in practice at most linear storage is allowed, we develop a simple approximation aIgorithm giving a seed set of size at most twice the size of the minimum. It requires O(n log2 n) time in 2D and 0(n2) time otherwise, and requires linear storage. We also give experimental results, showing the size of the seed sets and supporting the claim that sublinear storage is used.

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Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n log n) to O(n 13 ), and prove NP-hardness for some other variants.